dominate, have supremacy over

(しじょう) Being supreme, in the sense of #1 at something. #1 wrestler. #1 hot-dog eater. Also this is the word you use if you want to say white supremacist (白人 至上主義). Can also be used as an adjective, to mean supreme as in the  maximum of something. For example, the supreme happiness is 至上の喜び。Money makes you happy but only Jesus gives supreme happiness!!! And so on.
(はけん) Usually used about totalitarian governments. The other words in this cluster are often bad, but 覇権 is ALWAYS bad. Literally it translates to:a monopoly on power or rights. That should give you some idea.  Absolute dominance. No other political parties or rivals allowed.  It’s not limited to politics – any hegemony will do.   NP
(せいは)   also a political word, but not as judgmental as 覇権。 制覇 means to control something – an ocean, a bit of land. Usually used in history books:After the battle of 1645, Japan wrested control of the Japan Sea from Russia. Or – perhaps more usefully – about video game controls!
(どくせんてき) Usually translated as monopoly, but that’s too strong:monopoly means no one else is allowed to make the product. 独占 is more like coke, Pepsi, Microsoft:a company that dominates the marketplace.   $$$